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Visualisation in mathematics learning: canonical images and semiosis Tracy Wylie and Laurinda Brown Kingsfield School, South Gloucestershire and University of Bristol, GSoE, UK

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What is a canonical image? An image that is economical (Breen, 1997, PME 21) in that it gives direct access to the mathematical concept. What does this mean? Why might a teacher and researcher be interested in studying them?

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PME32 Proceedings Trigonometry has been discussed in short orals and a plenary: Plenary, Patrick Thompson, Vol 1 pp. 34-35 Focusing on misconceptions, SOs, Vol 1, p. 264 and p. 267 What is a canonical image for trigonometry?

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The trig image

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How would you do these? Students were asked to write down the value of (1/2 + √3/2) 5 Students were asked to find the general solution to the equation cos (2x + π/6) = 1/√2

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What happened? Hand gesture - Student F appears to be mimicking the shape of the cosine function. His pen is moving back and forth over the crest of the graph. It goes like that [gestures with hand] so it’s gonna be… no you’re gonna have to get the other one is 0 minus whatever the first one is because of the graph [draws on own sheet] ‘cause the graph will be like that and that point there is say π by 3, for instance, and that’s going to be minus…

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Where next? I began this piece of research with a canonical image for trigonometry. I would now suggest that an image is canonical if it can allow an individual working on mathematics to have the flexibility to use the image directly in a number of ways with a variety of problems.

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Trigonometry Section 7.4 Find the sine and cosine of special angles. Consider the angles 20 o and 160 o Note: sin 20 o = sin160 o and cos 20 o = -cos 160.

Trigonometry Section 7.4 Find the sine and cosine of special angles. Consider the angles 20 o and 160 o Note: sin 20 o = sin160 o and cos 20 o = -cos 160.

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